Mesoscopic elasticity controls dynamin-driven fission of lipid tubules

Mesoscale physics bridges the gap between the microscopic degrees of freedom of a system and its large-scale continuous behavior and highlights the role of a few key quantities in complex and multiscale phenomena, like dynamin-driven fission of lipid membranes. The dynamin protein wraps the neck formed during clathrin-mediated endocytosis, for instance, and constricts it until severing occurs. Although ubiquitous and fundamental for life, the cooperation between the GTP-consuming conformational changes within the protein and the full-scale response of the underlying lipid substrate is yet to be unraveled. In this work, we build an effective mesoscopic model from constriction to fission of lipid tubules based on continuum membrane elasticity and implicitly accounting for ratchet-like power strokes of dynamins. Localization of the fission event, the overall geometry, and the energy expenditure we predict comply with the major experimental findings. This bolsters the idea that a continuous picture emerges soon enough to relate dynamin polymerization length and membrane rigidity and tension with the optimal pathway to fission. We therefore suggest that dynamins found in in vivo processes may optimize their structure accordingly. Ultimately, we shed light on real-time conductance measurements available in literature and predict the fission time dependency on elastic parameters.

In order to construct a mesoscale interpretation of the dynamin action on axisymmetric lipid tubules, eventually defining p(r, z) of the main text, we ought to start from the structure of the polymer and the molecular level activities characterizing the GTP-consuming conformational changes.The prototypical dynamin monomer comprises five domains [1][2][3][4] organized as follows.A rigid stalk connects to the membrane-binding pleckstrin homology domain (PH) via flexible linkers and to the outwards-facing GTPase domain (G) through a bundle signaling element (BSE).The latter links to the stalk and G domains via the flexible joints hinge 1 and hinge 2, respectively, while an unstructured C-terminal, the proline-rich domain (PRD), interacts with endocytic recruiting proteins.Stalks are found to crisscross associate via interfaces 2 in an anti-parallel dimer, itself constituting the building block of the helix, and consequently tetramerize via interfaces 1 and 3 [3,5], these being associated to relatively soft bending modes [1,6].The geometry and associated mechanics of the resultant helix are therefore defined by these interfaces and oligomerization may only take place on compliant, high-curvature, and quasi-cylindrical lipid substrates.
Supposing the helix to be initially preassembled on the cylindrical substrate in the absence of GTP, we may model the dynamin polymer as a chain, where each link stands for one anti-parallel stalk dimer.The PH domains of each link are thence anchored to the underlying bilayer and are responsible for the force transmission onto the tubule [7].Concurrently, G domains of adjacent rungs face each other on the outer layer of this chain-like structure while bundle signaling elements (BSEs) flexibly connect them to the stalk links.Overall, these domains are coherently distributed in concentric cylinders [8], where the lipid-protein interfaces occupy the innermost layer, oppositely to G domains.Upon GTP addition, dimerization of apposed G domains is favored and, consequently, GTP hydrolysis triggers the conformational change consisting in an angular contraction between the BSE and G domains about the flexible hinge 2. Eventually, the hydrolyzed nucleotide is released, thus completing the cycle that determines the GTP-consuming, ratchet-like power stroke [6,[9][10][11].The latter mechanism results in an attraction force between the interested links of the adjacent rungs and, given the specific geometry of G domains cross-dimerization, translates into a force component parallel to the tubule axis, which is readily discharged on the tightly packed helix, and, more interestingly, a force component tangential to the cylindrical structure and normal to the tubule's axis.A joint effort of molecular dynamics, structure-based simulations [12] and single-molecule FRET observations provided a measure for this latter component, resulting in a tangential force F τ ≈ 2.5 to 4 pN acting so as to make the dimer slide in a helix-constriction sense [6].Noteworthy, the elastic response of the helix to changes in curvature and torsion throughout tubule constriction is found to be subdominant with respect to the energy cost for deforming the membrane [6].Altogether, these features substantiate the mesoscale interpretation of the dynamin polymer as a tightly packed helical chain coating the lipid tubule on a H-wide region, qualitatively depicted in Figure 1a of the main text and Figure S.1a here.Defining z the axial coordinate of the tubule with zero reference at the middle of the coat and r the radial distance from the latter axis, each element of the chain occupies a width comparable to the helical pitch h ≈ 10 nm in the z-direction, rationalized by the strict packing visible in Cryo-EM experiments combined with 3D map reconstructions [2,7,13,14], whilst having a ∼ 15 nm thickness on the orthogonal plane spanned by r.Due to the relatively large thickness of the dynamin coat, the external radii and pitch of the helix as well as the overall height H of the coat experience negligible deformations for our purposes [15,16].Furthermore, the discussed conformational changes come along with a tilting of the stalk [11], thus reducing the distance between consecutive motors and keeping unaltered the external geometry during constriction.As a matter of fact, the number of power stroke units per rung does not change sensibly during the whole constriction process and assesses on a value of N d ≈ 13 [2,3,5,10,13,15,16].
We introduce a helical curve following the ideal locations of the stalk dimers (black curve in Figure S.1a and b).This is defined by a h ≈ 10 nm pitch and has a R ≈ (r m + 8 nm) radial distance from the z-axis of the tubule, whose midsurface radius is r m .This curve is parametrized by its arc length s ∈ [0, L] and defines a Frenet-Serret orthonormal The dynamin polymer has a helical structure and is here represented with a (green) chain coiled around the (purple) lipid tubule in the initial, undeformed configuration.The black helical curve, parametrized by its arc length s, passes through the hypothetical locations of the stalk domains and defines the Frenet-Serret orthonormal basis shown in white.Each chain link stands for a dynamin dimer and, therefore, two oppositely oriented G-domains stick out from each.The G-domains of adjacent rungs interact in presence of GTP and produce an attractive force (the red arrow F ) between the interested links (one of which is shown in red).This force is statistically present in every pair of oppositely facing chain elements of adjacent rungs with the appropriate inclination, see [6] for a detailed explanation.(b) In order to model the pressure experienced by the lipid bilayer during dynamin constriction, we sketch the structure of a short (angular span is dθ and radius R) helix branch where the dynamin chain and the interested membrane region do balance.Under the hypotheses detailed in the text, each piece of the helical curve (black curve) is subjected to a tangential tension N (s) and a distributed tangential load fτ representing the diffuse action of the power strokes.Finally, the diffuse chain (shown in green) is balanced via a radial force per unit length exerted by the membrane reaction h pm and thought of as acting on its rm-radius midsurface.The assumption of quasi-cylindrical structure leads to n ≈ −r and b ≈ ẑ, with z and r the axial and radial coordinates of the axisymmetric system.
basis composed of the tangential τ , normal n, and binormal b unit vectors.The chain is able to bear axial tension (N (s) τ ), but no relevant bending response is expected in the range of deformations it experiences [1,6].Moreover, a mechanical balance of forces may be assumed since inertial and damping effects are negligible for the chain.Based on this, we can evaluate the axial load the chain experiences during constriction and to which the enclosed bilayer shall react through a radial pressure p m .Noticeably, only radial forces are expected from the bilayer since shear stresses due to viscous effects on the bilayer surface are not included in this analysis, following the remarked [17,18] time-scale separation between the fast lateral reorganization of lipids (∼ 10 ms) and the slow dynamin kinetics (∼ 0.1 to 1 s [1]).Under the hypotheses that the tubule is axisymmetric and quasi-cylindrical (i.e.r m varies slowly along z) and recognizing the modest inclination of the helix (h/(2πR) ≪ 1), we approximate each rung as horizontal and enforce n ≈ −r and b ≈ ẑ.All these considerations imply that R only slightly changes along the helix length.The force F exerted by the power stroke on a single link is depicted by the red arrow in Figure S.1a.The first homogenization step consists of distributing the tangential components F τ of these forces as a force per unit length along s, namely τ : We recall that the force component along ẑ is straightforwardly balanced by the strict packing of the chain.Since dθ is infinitesimal, p(s) and f τ (s) vary slowly along s, and ds = R dθ, at the first order of approximation we get Noticeably, in Equation S.1 the radial force stemming from the axial tension N (s) is balanced by the force per unit length sustained by the lipid bilayer (here thought of as exerted by the membrane midsurface).This latter consists of the pressure p m times the height of the contact region of each dimer, approximated as h.Since each chain link presents two oppositely oriented G domains, the inner turns of long polymers are statistically in balance.As a result, f τ (s) is different from zero only in the rungs at the edge of the helix and these effectively build up the tension that is thereof maintained constant in the inner branches of the chain.Altogether, we solve for the axial tension and membrane pressure as In these expressions, R ext indicates the radius of the (equal) external rungs of the helix and it is considered constant therein.Following the above assumptions and enforcing an axisymmetry approximation, it is possible to change the parametrization from the arc length s to the tubule axial coordinate z = s h/(2πR) − H/2 and radius r m .Eventually, we obtain the z-dependence shown in the red graph of Figure 1 and reading where h = min(h, H − h) takes into account the case of short dynamins with H < 2h.As reasonable, this expression is valid for polymers that present at least one full turn, i.e.H > h.
Since the dynamin helix cannot constrict down to a null internal lumen of the PH domains due to its molecular structure [5,13], we smoothly fade the pressure to zero between r m ≈ 3 nm and r m = 0, as evidenced in the blue graph of Figure 1c.The specific choice of the decaying function is not relevant as long as it is smooth and does not affect the pressure profile for r m > 3 nm.The modified mesoscopic pressure acting on the membrane (therefore oppositely oriented with respect to what shown in Figure S.1b) and entering the mathematical model in Equation 10of the main text is ultimately expressed in the cylindrical coordinates r and z as The dynamin is supposed to depolymerize and stop its constricting action whenever the tubule ruptures [19] and we implement this feature by annihilating the pressure field as soon as the value ϕ(r = 0, z) = − tanh(3/ √ 2) is reached anywhere on the axis, i.e. when even the outermost surface of the bilayer is severed.Noticeably, in our model, no depolymerization energy is deposited on the lipid membrane, in accordance with recent findings [10,20].

Lumen conductance
Among the available experimental techniques, measuring the conductance of the inner lumen of a tubule is a wellknown, convenient, and quite precise (in terms of geometry and time resolution) mean to follow real-time constriction of lipid tubules [19,21,22].In order to estimate the mobility of the evolution dynamics adopted in our numerical approach and, additionally, propose an explanation for dynamin-driven constriction data, we compute and measure the lumen conductance in our simulations.
There exist different levels of accuracy in computing the conductance of a channel.In particular, when the crosssectional dimensions of the system are in the order of a few nanometers, the accumulation of additional charge carriers in the Debye layer near the membrane walls influence the electric resistance of the lumen.Following the indications of [21][22][23], we compute the Debye corrected lumen conductance as (S.7) Here, D = 2 × 10 −9 m 2 s −1 is the ion diffusivity, e the elementary charge, C b = 0.15 M/L the bulk charges concentration, σ c = 0.5 e nm −2 the charge density on the inner monolayer, r the radius of the inner lumen, and L the length of the tubule.The first ration in the conductance expression is usually called bulk specific resistivity, The reported values are adopted from the indications of [19,21,22].This expression for the Debye-corrected conductance was proved to be effective for lumen radii down to ∼ 2 nm in the case of mean-field approaches with charged capillary walls [23,24].Eventually, the normalized conductance of the numerical tubule lumen is computed as with r 0 the initial lumen radius.

Determining the mobility and reference time of the system
Though most of the results discussed in this work pertain to time-independent observables of the system, such as critical or equilibrium quantities, we are interested in assessing the physical meaning of the dissipative dynamics imposed to the order parameter.Therefore, we both validate the consistency of Equation 14with real-time experimental observations as well as estimate the mobility of the system, M pf .The selected benchmark is the time-resolved normalized conductance of the tubule lumen during constriction induced by osmotic pressure, provided by Bashkirov et al. in [19].We reproduce the same setting by imposing γ = 5 × 10 −4 N m −1 (a value taking into account the electric potential effects on surface tension [22]), k b = 16 k B T , and R in ≈ 8 nm thereof.The tubule length is L = 1.4 µm and the external osmotic pressure is computed as with N A the Avogadro constant and l me the membrane thickness.The initial osmotic concentration inside the tubule is c 0 in and its difference with the external aqueous phase osmolarity is ∆c 0 at the beginning.As shown in [19], the pressure exerted on the lipid bilayer is proportional to the osmotic concentration difference but, as pointed out in [25], the inner environment becomes more concentrated the more the tubule is constricted, thereby providing the dependence on r in Equation S.9.By imposing osmotic concentrations analogous to those in [19], i.e. c 0 in = 0.22 Osm and ∆c 0 = −0.38Osm, we evolve the system and measure the Debye-corrected and normalized conductance of the lumen.Figure S.2 shows the comparison of the numerical and experimental conductances, G num n (blue dots) and G ex n (black shaded curve), respectively.From these, we are able to estimate the reference time of the numerical solution.In particular, we first fit the two curves with exponentially decaying functions with the same steady-state conductances and then match their decay times.Ultimately, we get the reference time τ R = 0.005 s for the numerical results.The non-dimensional form of Equation 14 provides a relationship between τ R and the phase field mobility, resulting in M pf = 4.04 nm 3 /(s k B T ).As will be demonstrated later on, this phase field mobility is, in general, linked to the sharp one, M sharp , by the specific choice of ϵ.

Estimating the analytical fission time
Under the hypothesis of perfectly cylindrical tubule and related dynamics, with R being the radius of its mid-surface Γ, it is possible to explicitly link the order parameter to the geometry of the membrane.In particular, defining n the signed distance from Γ, we write where n(t) = r − R(t) .(S.10) If the tubule radius varies by a small amount, δR, the corresponding change expected on the order parameter is On the other hand, we know that the free energy functional depending on the phase field, namely the Ginzburg-Landau free energy in Equation 3, converges to the sharp energy (Equation 1 augmented by a proper interaction contribution) in the limit of vanishing ϵ.It is worth noticing that we are referring to a condition where the Canham-Helfrich theory is expected to hold since the hypothesis of small width-to-radius ratio is satisfied.As also supported by [26], not only the diffuse interface energy converges to the sharp one, but also its variation as long as it is determined by a suitable displacement (i.e.following a gradient flow).Defining R in = k b /(2γ) and given a certain δR, then where the sharp free energy function, H sharp (R), consists in the Canham-Helfrich Hamiltonian augmented with an interaction term and only accounting for perfectly cylindrical configurations.In particular, the interaction is expressed by an inward pressure field, p(R), depending on the sole radius, R, of the cylindrical membrane.It is then possible to write the sharp energy as Ultimately, we retrieve the evolution equation for the tubule radius corresponding to the solution of the phase field dynamics in Equation S.15, reading Equation S.20 can also be straightforwardly obtained from the Allen-Cahn dynamics based on the general sharp energy, H sharp [u], and reads where which, in the case of free, unforced tubule (p(R) ≡ 0), is solved by For an unforced tubule with ρ ∼ 1 and ρ ̸ = 1, the solution is approximated by where the free relaxation time is This same approach might be exploited in order to get an estimate of the fission time in the presence of a dynamin covering the entire tubule.In this scenario, the pressure is p(R) = N d Fτ hR , and the solution gets Under the hypotheses of ρ ∼ 1 and ρ(t = 0) = 1, the approximate solution is Though it is valid only in the assumption of ρ ∼ 1 and fully-covering coat, Equation S.32 approximates well enough the dynamics of the central and neck radii obtained from the complete numerical simulations for long dynamins (H > H ⋆ ).Moreover, Equation S.32 provides the time for reaching ρ = 0, namely the characteristic fission time, which is expressed as  Fitted correction of the analytical fission time.Equation S.33 provides an analytical fission time for perfectly cylindrical systems.For actual cases, dynamins have a finite length, H, and membranes have a finite thickness, lme, requiring a correction to be introduced therefor.The numerical fission times found in the diffuse interface simulation campaign are shown as symbols for different elastic parameters.Noticeably, the numerical fission time does not depend on H when H > H ⋆ and the small, fitted correction only depends on Fτ .In the case of H < H ⋆ , the correction depends both on Rin and H.The solid lines show the fission time prediction modified by such corrections, see Equation S.36

FIG. S. 1 .
FIG. S.1.The mesoscopic effect of dynamin on the lipid tubule.(a)The dynamin polymer has a helical structure and is here represented with a (green) chain coiled around the (purple) lipid tubule in the initial, undeformed configuration.The black helical curve, parametrized by its arc length s, passes through the hypothetical locations of the stalk domains and defines the Frenet-Serret orthonormal basis shown in white.Each chain link stands for a dynamin dimer and, therefore, two oppositely oriented G-domains stick out from each.The G-domains of adjacent rungs interact in presence of GTP and produce an attractive force (the red arrow F ) between the interested links (one of which is shown in red).This force is statistically present in every pair of oppositely facing chain elements of adjacent rungs with the appropriate inclination, see[6] for a detailed explanation.(b) In order to model the pressure experienced by the lipid bilayer during dynamin constriction, we sketch the structure of a short (angular span is dθ and radius R) helix branch where the dynamin chain and the interested membrane region do balance.Under the hypotheses detailed in the text, each piece of the helical curve (black curve) is subjected to a tangential tension N (s) and a distributed tangential load fτ representing the diffuse action of the power strokes.Finally, the diffuse chain (shown in green) is balanced via a radial force per unit length exerted by the membrane reaction h pm and thought of as acting on its rm-radius midsurface.The assumption of quasi-cylindrical structure leads to n ≈ −r and b ≈ ẑ, with z and r the axial and radial coordinates of the axisymmetric system.
Figure S.1b, with N d the number of dimers per rung.The sketch in Figure S.1b assists in the expression of the mechanical balance for a short branch of the helix (or chain), where r m and R are essentially constant.Indicating with dθ the infinitesimal angular span of this branch and with ds its extension on the arc length, the balance reads r : −N (s) sin dθ 2 − N (s + ds) sin dθ 2 + dθ/2 −dθ/2 h p m (s) r m cos α dα = 0 , (S.1) FIG. S.2.Reference time estimation through osmotic pressure constriction.The Debye-corrected and normalized conductance of the tubule internal lumen is shown during the time evolution of a tubule constricted by a hypertonic solution.Numerical (blue dots, with numerical time t ⋆ on top axis) and experimental[19] (black shaded curve, with actual time on the bottom axis) results are shown.The same mesoscopic experimental conditions are reproduced in the simulation.Specifically, γ = 5 × 10 −4 N m −1 , k b = 16 kBT , and Rin ≈ 8 nm thereof.The osmotic pressure is computed as indicated by Equation S.9 in the text, with c 0 in = 0.22 Osm and ∆c 0 = −0.38Osm, the latter value granting the same steady state conductance.Exponentially decaying functions are used to fit the data (dashed black line for experimental and dashed red for numerical) in order to retrieve the ratio of the decay times, b/d = τR = 0.005 s.For the sake of completeness, the other fitting parameters are a ≈ b ≈ 0.51.